NONLINEAR VOLTERRA DIFFERENCE
EQUATIONS IN SPACE l p
MICHAEL I. GIL’ AND RIGOBERTO MEDINA
Received 4 December 2003
We consider a class of vector nonlinear discrete-time Volterra equations in space l p and
derive estimates for the norms of solutions. These estimates give us explicit stability conditions, which allow us to avoid finding Lyapunov functionals.
1. Introduction and statement of the main result
Volterra difference equations arise in the mathematical modeling of some real phenomena, and also in numerical schemes for solving differential and integral equations (cf.
[7, 8] and the references therein).
One of the basic methods in the theory of stability and boundedness of Volterra difference equations is the direct Lyapunov method (see [1, 3, 4] and the references therein).
But finding the Lyapunov functionals for Volterra difference equations is a difficult mathematical problem.
In this paper, we derive estimates for the c0 - and l p -norms of solutions for a class of
vector Volterra difference equations. These estimates give us explicit stability conditions.
To establish the solution estimates, we will interpret the Volterra equations with matrix
kernels as operator equations in appropriate spaces. Such an approach for Volterra difference equations has been used by Kolmanovskii and Myshkis [7], Kolmanovskii et al.
[8], Kwapisz [9], Medina [10, 11], and Gil’ and Medina [6]. Under some restriction, our
results generalize the main results from [6, 8, 11].
Let Cn be an n-dimensional complex Euclidean space with the Euclidean norm
· Cn . For a positive r ≤ ∞, put
ωr = h ∈ Cn : · Cn ≤ r .
(1.1)
As usual, c0 = c0 (Cn ) is the Banach space of sequences of vectors from Cn equipped with
the norm
hc0 = sup hk Cn
k
h = hk
∞
k=1
∈ c0 , hk ∈ Cn , k = 1,2,...
Copyright © 2004 Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society 2004:2 (2004) 301–306
2000 Mathematics Subject Classification: 39A10
URL: http://dx.doi.org/10.1155/S1026022604312021
(1.2)
302
Nonlinear Volterra difference equations in space l p
and l p = l p (Cn ) (1 < p < ∞) is the Banach space of sequences of vectors from Cn equipped
with the norm
h l p =
∞
p
h k n
k=1
1/ p
h = hk
,
C
∞
k=1
∈ l p , hk ∈ Cn , k = 1,2,... .
(1.3)
Let
a jk h1 ,...,h j −1
h1 ,...,h j −1 ∈ ωr ; k < j, j = 1,2,...
(1.4)
be n × n matrices dependent on j − 1 arguments. Consider the equation
x j = G j x1 ,...,x j −1 +
j −1
a jk x1 ,...,x j −1 xk
( j = 1,2,...),
(1.5)
k=1
where the mappings G j : C( j −1)n → Cn have the properties
f j :=
sup
h1 ,...,h j −1 ∈ωr
G j h1 ,...,h−1 Cn
<∞
( j = 1,2,...).
(1.6)
Moreover, G1 ∈ Cn is given and
f := f1 , f2 ,... ∈ l p (C).
(1.7)
v jk = sup a jk Cn < ∞ (k < j, j = 1,2,...),
(1.8)
In addition, it is assumed that
h1 ,...,h j −1

N p (V ) := 
∞
j −1
j =1
k=1
p
v jk
p/ p 1/ p
 <∞
(1.9)
with
1 1
+ = 1.
p p
(1.10)
M. I. Gil’ and R. Medina 303
To formulate the result, denote
m p (V ) =
∞
N pk (V )
√
,
p
k!
k=0
j −1
Q p (V ) = sup
j =1,2,...
k=1
p
v jk
(1.11)
1/ p
.
Now we are in a position to formulate the main result of the paper.
Theorem 1.1. Let conditions (1.7) and (1.9) hold. Then a solution x = (x1 ,x2 ,...) of (1.5)
satisfies the inequalities
xl p ≤ m p (V ) f l p ,
xc0 ≤ f c0 + Q p (V )m p (V ) f l p ,
(1.12)
f c0 + m p (V )Q p (V ) f l p < r.
(1.13)
provided
Note that due to the Hölder inequality,
∞
ak N pk (V )
√
≤
m p (V ) =
k p
k=0
a
k!
∞
1/ p a
k p
k=0
pk
∞
N p (V )
k=0
ak p k!
1/ p
(1.14)
for any positive a < 1. So
m p (V ) ≤ 1 − a
p −1/ p
p
N p (V )
exp
.
pa p
(1.15)
In particular, taking
a=
p
1
,
p
(1.16)
we have
p
m p (V ) ≤ b p exp N p (V ) ,
(1.17)
where
bp = 1 −
1
p p / p
−1/ p
.
(1.18)
304
Nonlinear Volterra difference equations in space l p
2. Proof of Theorem 1.1
First, assume that r = ∞. Then conditions (1.7) and (1.9) imply
x j Cn
≤ fj +
j −1
k=1
v jk xk Cn
( j = 1,2,...).
(2.1)
Define on l p = l p (R) the operator V by
[V h] j =
j −1
hk .
(2.2)
k=1
Here, [h] j means the jth coordinate of the element h ∈ l p (R). The operator V is a
quasinilpotent one. So, due to the well-known lemma from the book by Dalec’kiı̆ and
Kreı̆n (see [2, Lemma 3.2.1]) (the comparison principle),
x j Cn
≤ yj,
(2.3)
where y j is a solution of the equation
yj = fj +
j −1
v jk yk
( j = 1,2,...).
(2.4)
k=1
Rewrite this equation as
y = f + V y.
(2.5)
Lemma 2.1. Let conditions (1.7) and (1.9) hold. Then a solution y of (2.5) satisfies the
inequality
y l p ≤ m p (V ) f l p .
(2.6)
y = (I − V )−1 f .
(2.7)
N pk (V )
k
V p ≤ √
.
p
l
(2.8)
Proof. Rewrite (2.5) as
By [5, Lemma 4.3],
k!
Since
(I − V )−1 =
∞
k=0
V k,
(2.9)
M. I. Gil’ and R. Medina 305
we have
(I − V )−1 p ≤ m p (V ),
l
(2.10)
concluding the proof.
Lemma 2.2. Let conditions (1.7) and (1.9) hold. Then a solution y of (2.5) satisfies the
inequality
y c0 ≤ f c0 + m p (V ) f l p .
(2.11)
Proof. From (2.5) it follows that
y c0 ≤ f c0 + V y c0 .
(2.12)
But due to Hölder’s inequality
V y c0 ≤ sup
j =1,2,...
j −1
k=1
p
v jk
1/ p
y l p = Q p (V ) y l p ,
(2.13)
now (2.12) and Lemma 2.1 yield
y c0 ≤ f c0 + Q p (V ) y l p ≤ f c0 + Q p (V )m p (V ) f l p
as claimed.
(2.14)
Proof of Theorem 1.1. If r = ∞, then the required result follows from Lemmas 2.1 and 2.2.
Let now r < ∞. By a simple application of the Urysohn lemma and Lemma 2.2, we get the
required result.
Acknowledgment
This research was supported by Kamea fund of Israel and by Fondecyt (Chile) under
Grant no. 1.030.460.
References
[1]
[2]
[3]
[4]
[5]
[6]
M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, Stability of continuous and discrete
Volterra integro-differential equations by Liapunov approach, J. Integral Equations Appl. 7
(1995), no. 4, 393–411.
J. L. Dalec’kiı̆ and M. G. Kreı̆n, Stability of Solutions of Differential Equations in Banach Space,
American Mathematical Society, Rhode Island, 1974.
S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics,
Springer-Verlag, New York, 1996.
S. N. Elaydi and S. Murakami, Asymptotic stability versus exponential stability in linear Volterra
difference equations of convolution type, J. Differ. Equations Appl. 2 (1996), no. 4, 401–410.
M. I. Gil’, Invertibility and spectrum of Hille-Tamarkin matrices, Math. Nachr. 244 (2002), 78–
88.
M. I. Gil’ and R. Medina, Boundedness of solutions of matrix nonlinear Volterra difference equations, Discrete Dyn. Nat. Soc. 7 (2002), no. 1, 19–22.
306
[7]
[8]
[9]
[10]
[11]
Nonlinear Volterra difference equations in space l p
V. B. Kolmanovskii and A. D. Myshkis, Stability in the first approximation of some Volterra difference equations, J. Differ. Equations Appl. 3 (1998), no. 5-6, 563–569.
V. B. Kolmanovskii, A. D. Myshkis, and J.-P. Richard, Estimate of solutions for some Volterra
difference equations, Nonlinear Anal. Ser. A: Theory Methods 40 (2000), no. 1–8, 345–363.
M. Kwapisz, On l p solutions of discrete Volterra equations, Aequationes Math. 43 (1992), no. 2-3,
191–197.
R. Medina, Solvability of discrete Volterra equations in weighted spaces, Dynam. Systems Appl. 5
(1996), no. 3, 407–421.
, Stability results for nonlinear difference equations, Nonlinear Stud. 6 (1999), no. 1, 73–
83.
Michael I. Gil’: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653,
Beer-Sheva 84105, Israel
E-mail address: gilmi@cs.bgu.ac.il
Rigoberto Medina: Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933,
Osorno, Chile
E-mail address: rmedina@ulagos.cl